Experiment-03

3.1 Title: Analysis of PCM and DPCM Systems and Interpretation of Modulated and Demodulated Waveforms

3.2 Aim of the experiment:
To analyze the operation of Pulse Code Modulation (PCM) and Differential Pulse Code Modulation (DPCM) systems, interpret their modulated and demodulated waveforms, and investigate the magnitude spectrum of PCM signals for sampling frequencies of 4 kHz and 8 kHz

3.3 Theoretical background for the experiment:

Pulse Code Modulation (PCM): PCM is a digital modulation technique in which the analog signal is sampled, quantized, and encoded into a digital bit stream. The process involves:

1. Sampling: Converting a continuous-time signal into a discrete-time signal.

2. Quantization: Approximating sampled values to the nearest value within a finite set.

3. Encoding: Representing quantized values using binary codes.

Differential Pulse Code Modulation (DPCM): DPCM is an extension of PCM where only the difference between consecutive samples is quantized and encoded, reducing redundancy and improving bandwidth efficiency.

Key Equations:

1. Sampling Frequency: Where is the sampling frequency and is the maximum frequency of the analog signal.

2. Quantization Error: Where is the original sample value and is the quantized value.

3.4 Design:

1. Generate an analog sinusoidal signal as the input.

2. Implement PCM by sampling, quantizing, and encoding the signal.

3. Implement DPCM by encoding the difference between consecutive samples.

4. Perform demodulation for both PCM and DPCM systems to reconstruct the original signal.

3.5 Step-by-step procedure to carry out the experiment:

1. Define an analog input signal (e.g., a sine wave of 1 kHz frequency).

2. Sample the input signal at 4 kHz and 8 kHz.

3. Perform uniform quantization to encode the sampled values into PCM.

4. Calculate differences between consecutive samples for DPCM encoding.

5. Decode PCM and DPCM signals to reconstruct the original signal.

6. Compute the Fourier Transform of the PCM signal to obtain its magnitude spectrum.

7. Compare the original signal with the demodulated signals for accuracy.

3.6 Code:
Pulse Code Modulation

% Parameters

Fs = 1000;        % Sampling frequency (Hz)

t = 0:1/Fs:1-1/Fs; % Time vector (1 second)

f = 5;           % Signal frequency (Hz)

A = 1;           % Signal amplitude

% Generate a sinusoidal signal

signal = A * sin(2 * pi * f * t);

% Plot the original signal

figure;

subplot(3, 1, 1);

plot(t, signal, 'b');

title('Original Signal');

xlabel('Time (s)');

ylabel('Amplitude');

grid on;

% PCM Process

n_bits = 3; % Number of quantization levels (3 bits -> 2^3 = 8 levels)

% Normalize the signal to the range [0, 1]

signal_norm = (signal - min(signal)) / (max(signal) - min(signal));

% Quantize the signal

quant_levels = 2^n_bits;

quantized_signal = round(signal_norm * (quant_levels - 1));

% Plot the quantized signal

subplot(3, 1, 2);

stairs(t, quantized_signal, 'g');

title('Quantized Signal');

xlabel('Time (s)');

ylabel('Quantized Level');

grid on;

% Convert to binary (PCM encoding)

pcm_encoded = dec2bin(quantized_signal, n_bits); % Convert to binary matrix

% Display a sample of binary values for verification

disp('Sample PCM Encoded Values (Binary):');

disp(pcm_encoded(1:10, :)); % Display first 10 encoded samples

% PCM Decoding (Reconstruction)

decoded_signal = bin2dec(pcm_encoded) / (quant_levels - 1); % Convert back to decimal

decoded_signal = decoded_signal * (max(signal) - min(signal)) + min(signal); % Denormalize

% Plot the decoded signal

subplot(3, 1, 3);

plot(t, decoded_signal, 'r');

title('Decoded Signal');

xlabel('Time (s)');

ylabel('Amplitude');

grid on;

% Superimpose original and decoded signals for comparison

figure;

plot(t, signal, 'k', t, decoded_signal, 'r--');

legend('Original Signal', 'Decoded Signal');

title('Comparison of Original and Decoded Signals');

xlabel('Time (s)');

ylabel('Amplitude');

grid on;

DPCM

% DPCM Implementation in MATLAB

% Step 1: Signal generation

fs = 1000; % Sampling frequency (Hz)

t = 0:1/fs:1; % Time vector (1 second duration)

f = 5; % Frequency of the signal (Hz)

signal = sin(2*pi*f*t); % Original signal

% Step 2: Quantization parameters

nBits = 4; % Number of bits for quantization

nLevels = 2^nBits; % Number of quantization levels

signal_min = min(signal);

signal_max = max(signal);

qLevels = linspace(signal_min, signal_max, nLevels); % Quantization levels

% Step 3: DPCM Encoding

encodedSignal = zeros(size(signal)); % Encoded signal (quantized differences)

prediction = 0; % Initialize prediction

quantizedIndices = zeros(size(signal)); % Store quantized indices for binary representation

for i = 1:length(signal)

% Compute the difference

difference = signal(i) - prediction;

% Quantize the difference

[~, idx] = min(abs(qLevels - difference));

quantizedDifference = qLevels(idx);

% Save the encoded difference

encodedSignal(i) = quantizedDifference;

quantizedIndices(i) = idx - 1; % Store the index (0-based

% Update the prediction (reconstruction)

prediction = prediction + quantizedDifference;

end

% Convert quantized differences to binary (DPCM encoded)

dpcmCode = dec2bin(quantizedIndices, nBits); % Binary encoding

% Step 4: DPCM Decoding

decodedSignal = zeros(size(encodedSignal));

prediction = 0; % Initialize prediction for decoding

for i = 1:length(encodedSignal)

% Reconstruct the signal

decodedSignal(i) = prediction + encodedSignal(i);

prediction = decodedSignal(i); % Update the prediction

end

% Step 5: Visualization

figure;

% Original signal

subplot(3, 1, 1);

plot(t, signal);

title('Original Signal');

xlabel('Time (s)');

ylabel('Amplitude');

% Encoded DPCM Signal (quantized indices)

subplot(3, 1, 2);

stairs(quantizedIndices,'g');

title('DPCM Encoded Signal (Quantized Indices)');

xlabel('Sample Index');

ylabel('Index');

% Reconstructed signal

subplot(3, 1, 3);

plot(t, decodedSignal);

title('Reconstructed Signal (DPCM Decoded)');

xlabel('Time (s)');

ylabel('Amplitude');

disp('DPCM Encoded Binary Data:');

disp(dpcmCode);

3.7 Observations and results:

1. The PCM signal represents the sampled and quantized version of the analog input.

2. The DPCM signal effectively reduces redundancy by encoding differences between samples.

3. The demodulated PCM and DPCM signals closely resemble the original analog signal.

4. The magnitude spectrum of the PCM signal shows discrete frequency components.

3.8 Plotting of the graph:

PCM

DPCM

3.9 Conclusion of the experiment: The PCM and DPCM system experiment using MATLAB successfully demonstrated the conversion of analog signals to digital form through quantization and encoding techniques. It highlighted how DPCM improves compression efficiency by encoding the difference between successive samples, reducing the bit rate.